A multiple of any quantity, is that which is some exact number of times that quantity. Thus, 12 is a multiple of 4, 15a is a multiple of 3a, and 20a2b3 of 5ab. The reciprocal of any quantity, is that quantity inverted, or unity divided by it. Thus, the reciprocal of a, or, is; and the reciprocal A function of one or more quantities, is an expression into which those quantities enter, in any manner whatever, either combined, or not, with known quantities. - Thus, a — 2x, ax+3x2, 2x — a (a3 —x2)1, axTM, aa, &c., are functions of x; and axy+bx2, ay+x (ax2—by2)1⁄2, &c. are functions of x and y. A vinculum, is a bar or parenthesis ( ), made use of to collect several quantities into one. Thus, a+bxc, or (a+b) c, denotes that the compound quantity a+b is to be multiplied by the simple quantity c; and ab+c2, or (ab+c2), is the square root of the compound quantity ab+c2. Practical Examples for computing the numeral Values of various Algebraic Expressions, or Combinations of Letters. Supposing a 6, 6=5, c=4, d=1, and e=0. Then 1. a2+2ab-c+d=36+60—4+1=93. 6. √2a2 −√/2ac+c3 − √√72−√✓/64=√72-8=√✓/64= 8. 2a+3c 4bc 12+12 6d+-4e ✔2ac+ca 6+0 80 6 48+16 24 80 8 Required the numeral values of the following quantities; supposing a, b, c, d, e, to be 6, 5, 4, 1, and 0, respectively, as above. 1. 2a+3bc-5d=127 2. 5a2b-10ab2+2e=-600 3. 7a2+b-cxd+e=253 4. 5√ab+b22ab-e2 = 7.613875 6. 3/c+2a√2a+b-d=14 7. ava2+b2+3bc✔✔a2 - b2=245.8589862 8. 3a2b+3c2+2ac+c2=542.8844991 26+c √56+3vc+d 9. ADDITION. ADDTION is the connecting of quantities together by means of their proper signs, and incorporating such as are like, or that can be united, into one sum; the rule for performing which is commonly divided into the three following cases* : CASE I. When the Quantities are like, and have like signs. RULE. Add all the coefficients of the several quantities together, and to their sum annex the letter or letters belonging to each term, prefixing, when necessary, the common sign. * The term Addition, which is generally used to denote this rule, is too scan. ty to express the nature of the operations that are to be performed in it; which are sometimes those of addition, and sometimes subtraction, according as the quantities are negative or positive. It should, therefore, be called by some name signifying incorporation, or striking a balance; in which case, the incon gruity, here mentioned, would be removed. When the Quantities are like, but have unlike signs. RULE. Add all the affirmative coefficients into one sum, and those that are negative into another, when there are se veral of the same kind; then subtract the least of these sums from the greatest, and to the difference prefix the sign of the greater, annexing the common letter or letters as before. CASE III. When the Quantities are unlike; or some like and others un like. RULE. Collect all the like quantities together, by taking their sums or differences, as in the foregoing cases, and set down those that are unlike, one after another, with their proper signs. -xy 4xy - axa +3αx3 x2+xy 4x2 6x2+xy 8α2x2 -3αx 9x Y - 5xy 5ax 5x2-3x2 3√x+10 8x2-20 1062-3a2 x a2x2+120 10a2x2+5xy-ax 962+3a2x2-a2x+170 -2xy2 3xy+10x |